Question

1. match the perfect square trinomials on the left to their factorization on the right.
2. $9x^8y^2z^{12} + 6x^8y^3z^7 + x^8y^4z^2$
3. $25x^8y^2z^{12} - 80x^8y^3z^7 + 64x^8y^4z^2$
4. $16x^6y^8z^4 + 16x^4y^7z^7 + 4x^2y^6z^{10}$
5. $49x^6y^8z^4 - 84x^4y^7z^7 + 36x^2y^6z^{10}$

a. $(5x^4yz^6 - 8x^4y^2z)^2$
b. $(4x^3y^4z^2 + 2xy^3z^5)^2$
c. $(3x^4yz^6 + x^4y^2z)^2$
d. $(7x^3y^4z^2 - 6xy^3z^5)^2$

Explanation:

Step 1: Recall the perfect square trinomial formula

A perfect square trinomial has the form \(a^{2}+2ab + b^{2}=(a + b)^{2}\) or \(a^{2}-2ab + b^{2}=(a - b)^{2}\). We analyze each trinomial by identifying \(a^{2}\), \(2ab\) (or \(-2ab\)), and \(b^{2}\).

Step 2: Analyze Trinomial 1

Trinomial 1: \(9x^{8}y^{2}z^{12}+6x^{8}y^{3}z^{7}+x^{8}y^{4}z^{2}\)

• \(a^{2}=9x^{8}y^{2}z^{12}\), so \(a = 3x^{4}yz^{6}\) (since \((3x^{4}yz^{6})^{2}=9x^{8}y^{2}z^{12}\)).
• \(b^{2}=x^{8}y^{4}z^{2}\), so \(b = x^{4}y^{2}z\) (since \((x^{4}y^{2}z)^{2}=x^{8}y^{4}z^{2}\)).
• Check \(2ab\): \(2(3x^{4}yz^{6})(x^{4}y^{2}z)=6x^{8}y^{3}z^{7}\), which matches the middle term. Thus, it factors to \((3x^{4}yz^{6}+x^{4}y^{2}z)^{2}\) (option c).

Step 3: Analyze Trinomial 2

Trinomial 2: \(25x^{8}y^{2}z^{12}-80x^{8}y^{3}z^{7}+64x^{8}y^{4}z^{2}\)

• \(a^{2}=25x^{8}y^{2}z^{12}\), so \(a = 5x^{4}yz^{6}\) (since \((5x^{4}yz^{6})^{2}=25x^{8}y^{2}z^{12}\)).
• \(b^{2}=64x^{8}y^{4}z^{2}\), so \(b = 8x^{4}y^{2}z\) (since \((8x^{4}y^{2}z)^{2}=64x^{8}y^{4}z^{2}\)).
• Check \(-2ab\): \(-2(5x^{4}yz^{6})(8x^{4}y^{2}z)=-80x^{8}y^{3}z^{7}\), which matches the middle term. Thus, it factors to \((5x^{4}yz^{6}-8x^{4}y^{2}z)^{2}\) (option a).

Step 4: Analyze Trinomial 3

Trinomial 3: \(16x^{6}y^{8}z^{4}+16x^{4}y^{7}z^{7}+4x^{2}y^{6}z^{10}\)

• \(a^{2}=16x^{6}y^{8}z^{4}\), so \(a = 4x^{3}y^{4}z^{2}\) (since \((4x^{3}y^{4}z^{2})^{2}=16x^{6}y^{8}z^{4}\)).
• \(b^{2}=4x^{2}y^{6}z^{10}\), so \(b = 2xy^{3}z^{5}\) (since \((2xy^{3}z^{5})^{2}=4x^{2}y^{6}z^{10}\)).
• Check \(2ab\): \(2(4x^{3}y^{4}z^{2})(2xy^{3}z^{5})=16x^{4}y^{7}z^{7}\), which matches the middle term. Thus, it factors to \((4x^{3}y^{4}z^{2}+2xy^{3}z^{5})^{2}\) (option b).

Step 5: Analyze Trinomial 4

Trinomial 4: \(49x^{6}y^{8}z^{4}-84x^{4}y^{7}z^{7}+36x^{2}y^{6}z^{10}\)

• \(a^{2}=49x^{6}y^{8}z^{4}\), so \(a = 7x^{3}y^{4}z^{2}\) (since \((7x^{3}y^{4}z^{2})^{2}=49x^{6}y^{8}z^{4}\)).
• \(b^{2}=36x^{2}y^{6}z^{10}\), so \(b = 6xy^{3}z^{5}\) (since \((6xy^{3}z^{5})^{2}=36x^{2}y^{6}z^{10}\)).
• Check \(-2ab\): \(-2(7x^{3}y^{4}z^{2})(6xy^{3}z^{5})=-84x^{4}y^{7}z^{7}\), which matches the middle term. Thus, it factors to \((7x^{3}y^{4}z^{2}-6xy^{3}z^{5})^{2}\) (option d).

Answer:

1. \(9x^{8}y^{2}z^{12}+6x^{8}y^{3}z^{7}+x^{8}y^{4}z^{2}\) matches with c. \((3x^{4}yz^{6}+x^{4}y^{2}z)^{2}\)
2. \(25x^{8}y^{2}z^{12}-80x^{8}y^{3}z^{7}+64x^{8}y^{4}z^{2}\) matches with a. \((5x^{4}yz^{6}-8x^{4}y^{2}z)^{2}\)
3. \(16x^{6}y^{8}z^{4}+16x^{4}y^{7}z^{7}+4x^{2}y^{6}z^{10}\) matches with b. \((4x^{3}y^{4}z^{2}+2xy^{3}z^{5})^{2}\)
4. \(49x^{6}y^{8}z^{4}-84x^{4}y^{7}z^{7}+36x^{2}y^{6}z^{10}\) matches with d. \((7x^{3}y^{4}z^{2}-6xy^{3}z^{5})^{2}\)